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Usually when a card is banned, I can figure out what the reason is, but for the life of me I can't figure out why this card is so dangerous. Its ability is good but not game-breaking, and I can't think of anything that it would pair with to become game-breaking.

4 Answers
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Basically it's because it buffers you very, very well against creature loss. If you're playing a weenie deck it protects you from deck-clearing spells. Or if you're sacrificing creatures, this card gives you extremely cheap, essentially unlimited new draws. The equip ability means you can just reuse it over and over again.

It's also very wide-ranging - almost any competitive deck is going to be better off having a few of these around, so it distorts the format. The basis for that statement is discussed in this very detailed article from the development team:

Skullclamp was banned in Standard, frankly, because it was everywhere.
Every competitive deck either had four in the main deck, had four in
the sideboard, or was built to try and defend against it. And there
were a lot more successful decks in the first two categories than in
the third. Such representation is completely unhealthy for the format.
Your deck has to either have Skullclamps, or have Skullclamp in its
crosshairs—a definitive case of a card “warping the metagame.”
Look, for example, at the Top 8 decks from Ohio Valley Regionals. Or
at those from the more recent German Nationals. Combined, those 16
decks contained 58 out of a possible 64 Skullclamps. Never in my
memory have I ever seen a card show up in those numbers.

It's interesting to note that the development team themselves completely underestimated the power of this card. The story of how it made it to release, discussed in the linked article, is quite fascinating.

I know this is an ancient post but I stumbled upon it in a Google search and figured I could add a bit more historical context.

As others have mentioned (and linked), Skullclamp was proven pernicious in the Standard and Extended formats of its day. There was actually a period where it seemed it was getting banned from a new format every B&R announcement (first Standard, then Extended, then Magic Online-only formats).

That said, nobody knows for certain what impact it would have on Legacy or Modern because it was never legal in either format. Both of those formats are relatively new, and unusual in that they're popular tournament formats that are younger than (most of) their card pools. Because both formats started with a large card base, both formats also started with banlists that are somewhat speculative. It's tricky business because if they started with nothing banned, both formats would have been ruled by known broken decks for quite some time (what Zvi Mowshowitz termed "Oh Lord, Not Again"), so Wizards extrapolated from past tournaments in other, similar formats.

The result is that both formats have a number of cards that were "preemptively" banned in this fashion. Sometimes those cards get unbanned; Mind over Matter, Replenish, and Land Tax are apparently not as bad as they were in their original contexts. I doubt the same thing can be said for Jace, The Mind Sculptor/Stoneforge Mystic in Modern, or Mind's Desire/Yawgmoth's Bargain in Legacy.

While Skullclamp was never proven broken in current formats as such, it ruined Standard, eventually proved broken in Extended and was almost Vintage playable. It's also arguably a more resilient Glimpse of Nature (and certainly has a similar effect in the same decks), and that card is also illegal in Modern. It's almost certainly staying banned forever in Modern, and deservedly so, since that format is intentionally kept at a power level at or below the old Extended formats where we already established Skullclamp was too good. Legacy is fast enough that I can see an argument for unbanning it coming up at some point in the far future, but there's a danger of it becoming an automatic 4-of in every aggro deck and becoming an autowin against control. There's also the risk that it fits too well in a creature-based combo deck like Elves!, or that it combos too well with some other cards printed in the many years since it was ever legal in a non-Vintage format (Stoneforge Mystic? Puresteel Paladin? Scars block in general?) So I don't like its chances there either.

[^1] Technically there was one Magic Online "Modern" tournament, which I believe was open only to employees, where Jace was legal. But then the standard bannings happened and by the time the Modern banlist was finalized for the general population Jace was on it.

EDIT: After some more googling I dug up the development articles from when the banlists were new: Aaron Forsythe on Legacy and Tom LaPille on Modern. It's telling that neither felt the need to go into much detail justifying the banning. Its effect on Standard and Extended really was that bad.

On the surface of things this may not look obviously broken, but honestly, in the right deck it's an engine that says "pay 1 mana: draw 2 cards". As the article points out, at the time all the top decks were playing 4 Clamps; much as Jace, The Mind Sculptor just got banned for being ubiquitous at recent Magic tournaments. When a card becomes a non-optional 4-of constituent of any competitive deck that hopes to win games, it has to go.

Then the card was considered being OK and nobody changed anything.. until one day they decided to improve some of the equipment, as it was one of the features of the block. It was then changed to what was actually published and, because this change has been made at the very last minute, it wasn't tested before the actual print; also it was not considered worth testing because the developers had in mind the old version of this card, which was pretty bad.

Then, although the R&D quickly understood that the card was broken, people didn't realize immediately how powerful it was, so Skullclamp wasn't banned at the first ban list possible; after that, the meta had evolved in pro-Clamp or anti-Clamp decks and the ban in Standard and Block-Constructed was needed. It took a little more time but then it was also banned in Legacy.